If REF  fails, then the \(U\) -stabilisers have non-constant dimension on \(Y_{\min }^{ss}\) , so \(d_{\min } < d_{\max }\) . In this case, we perform a sequence of \(H\) -equivariant blow-ups to arrange for REF  to hold on the blow-up. The idea is to start by considering points with maximal dimensional \(U\) -stabilisers. In fact, there are several possible ways to perform this blow-up sequence: one can blow-up the points in \(Y_{\min }\)  whose stabilisers have maximal \(U\) -dimension or blow-up the \(U\) -sweep of the locus of points in \(Z_{\min }\)  whose stabilisers have maximal \(U\) -dimension (the latter is equivalent to blowing-up the locus of points in \(Y_{\min }\)  with maximal dimensional \(\widehat{U}\) -stabiliser). In [1]}, the former approach is taken, but we will take the latter approach. This approach involves blowing up a smaller locus and it is easier to show this process terminates, as one is always blowing up a locus of codimension at least 2.
